203ch18_slides.pdf

# 203ch18_slides.pdf - STAT 203 Chapter 18 Inferences About...

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STAT 203 Chapter 18 Inferences About Means Recall that we use sample statistics to estimate population parameters. Since the value of a statistic varies from sample to sample, we can use the standard deviation to measure the variability of a statistic. Population parameter Sample statistic Standard deviation of statistic Standard error of statistic Population proportion p Sample proportion ˆ p SD p ) = p (1 - p ) n SE p ) = ˆ p (1 - ˆ p ) n Population mean μ Sample mean y SD ( y ) = σ n SE ( y ) = s n where σ and s are the population standard deviation and the sample standard deviation, respectively. Eugenia Yu, UBC Department of Statistics. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1

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STAT 203 Standard error of the sample mean y : In most situations, the value of the population standard deviation σ is unknown, and hence so is SD ( y ) = σ n . We will estimate σ using the sample standard deviation s from a random sample, and estimate SD ( y ) = σ n by the standard error SE ( y ) = s n . In Chapter 15, the confidence interval for a population proportion p is given by: ˆ p ± z * SE p ) where the critical value z * is based on the standard Normal model. In general, the confidence interval for a population parameter takes the following form: statistic ± critical value × (standard error of statistic) Eugenia Yu, UBC Department of Statistics. Not to be copied, used, or revised without explicit written permission from the copyright owner. 2
STAT 203 Confidence intervals for the population mean μ When σ is unknown , we’ll use SE ( y ) = s n , and the confidence interval for μ will be based on the t - model instead of the standard Normal model (see Appendix for the theoretical details). A confidence interval of confidence level C for μ is constructed using: y ± t * n - 1 s n where the critical value t * n - 1 can be obtained from the t-table. (See the t-table posted on the course website.) Such a confidence interval for μ is called the one-sample t -interval . Eugenia Yu, UBC Department of Statistics. Not to be copied, used, or revised without explicit written permission from the copyright owner. 3

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STAT 203 Properties of the t -model * perfectly symmetric about the mean 0 * unimodal, bell-shaped * has one model parameter – the degrees of freedom ( df ) which determines the shape of the model and is given by n - 1 * has thicker tails when sample size is smaller *

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